Gates have four kinds of connections:

 * Input
 * Output
 * Power
 * Ground

When you make a logic circuit using ICs, the first thing you need is to make sure they have their +5 and GND connections tied.

Once you've done that, connect your inputs and your outputs to the gate inputs and outputs.

Finally, if you have a gate output feeding another gate input, it's perfectly fine to tie them output-to-input. And, a single gate output can drive multiple outputs just by tying them directly.